Bounding the sum of powers of the Laplacian eigenvalues of graphs

For a non-zero real number α, let s _α (G) denote the sum of the αth power of the non-zero Laplacian eigenvalues of a graph G. In this paper, we establish a connection between s _α (G) and the first Zagreb index in which the Hölder’s inequality plays a key role. By using this result, we present a lot of bounds of s _α (G) for a connected (molecular) graph G in terms of its number of vertices (atoms) and edges (bonds). We also present other two bounds for s _α (G) in terms of connectivity and chromatic number respectively, which generalize those results of Zhou and Trinajsti? for the Kirchhoff index B Zhou, N Trinajsti?. A note on Kirchhoff index, Chem. Phys. Lett., 2008, 455: 120–123].